This paper is meant to give a brief survey on Quillen conjecture and finally present a recent result that this conjecture has been verified by the authors.
Trang 1Science & Technology Development Journal, 22(2):235- 238
Review
Faculty of Mathematics and Computer
Science University of Science,
VNU-HCMC
Correspondence
Bui Anh Tuan, Faculty of Mathematics
and Computer Science University of
Science, VNU-HCMC
Email: batuan@hcmus.edu.vn
History
•Received: 2018-12-05
•Accepted: 2019-03-31
•Published: 2019-06-14
DOI :
https://doi.org/10.32508/stdj.v22i2.1229
Copyright
© VNU-HCM Press This is an
open-access article distributed under the
terms of the Creative Commons
Attribution 4.0 International license.
A brief introduction to Quillen conjecture
ABSTRACT Introduction: In 1971, Quillen stated a conjecture that on cohomology of arithmetic groups, a
certain module structure over the Chern classes of the containing general linear group is free Over time, many efforts has been dedicated into this conjecture Some verified its correctness, some disproved it So, the original Quillens conjecture is not correct However, this conjecture still has great impacts on the field cohomology of group, especially on cohomology of arithmetic groups This paper is meant to give a brief survey on Quillen conjecture and finally present a recent result
that this conjecture has been verified by the authors Method: In this work, we investigate the
key reasons that makes Quillen conjecture fails We review two of the reasons: 1) the injectivity
of the restriction map; 2) the non-free of the image of the Quillen homomorphism Those two
reasons play important roles in the study of the correctness of Quillen conjecture Results: In
sec-tion 4, we present the cohomology of ring H ∗(
SL2
( Z[√ −2][1
2
])
;F2
) which is isomorphic to the free moduleF2[e4] (x2, x3, y3, z3, s3, x4, s4, s5, s6)overF2[e4] This confirms the Quillen conjecture
Conclusion: The scope of the conjecture is not correct in Quillens original statement It has been
disproved in many examples and also been proved in many cases Then determining the conjec-tures correct range of validity still in need The result in section 4 is one of the confirmation of the validity of the conjecture
Key words: Quillen conjecture, Cohomology of group, Arithmetic groups, S-arithmetic groups,
Cohomology ring
PRELIMINARIES
In this section, we will review some concepts related
to Quillen conjecture and then give a full and details
of the conjecture The key objects in the conjecture are
arithmetic and S-arithmetic groups whose concepts
arose when studying modular form and some other classical topics in number theory However, the defi-nition of arithmetic groups is very technical We can, roughly, think of arithmetic groups as the intersection
of SL(n, Z) (or GL(n,Z)) with some semi-simple Lie subgroup G of SL(n,R) (or GL(n,Z)) Full definition
can be seen in1as follows:
Definition 1 Let G be a semi-simple subgroup of
SL(n, R) Group Γ is said to be an arithmetic subgroup
of G if and only if Γ is a lattice in G and there exist a closed, connected, semisimple subgroup G ′of some
SL(n, R), such that G ′is defined overQ,
• a closed, connected, semisimple subgroup G ′of
some SL(n,R), such that G ′is defined overQ,
• compact normal subgroups K and K ′ of G ◦and
G ′, respectively, and
• an isomorphismψ : G ◦ /K → G ′ /K ′, such that
ψ(Γ) is commensurable to G ′
Z, whereΓ and
G ′
Zare images ofΓ ∩ G ◦ and G ′
Zin G ◦ /Kand
G ′ /K ′, respectively.
Reader can find definition of defined over, commensu-rable and other related concepts in2 Some common
arithmetic groups are SL n(Z),PSLn(Z),PGLn(Z)
modular groups, Siegel groups, etc.
In order to generalize the concept of arithmetic groups, one natural method is replacing the ringZ
of integers by a bigger ring There are two ways that lead to two different branches One is considering the ring of integers of some number fields, this leads to the concepts of Hilbert modular and Bianchi groups Another is replacingZ by the ring of integers with some primes invertedZ[1
p1, p1
2,··· , 1
p n
] The fusion
of these two ideas leads us to the definitions of S-integers and S-arithmetic groups.
Recall that given an algebraic number field K, an ab-solute value on K is a map |.| v : K → R ≥0satisfies
• | α| v= 0if and only ifα = 0,
• | αβ| v=| α| v | β| vfor allα,β ∈ K,
• there exists a > 0 such that | α + β| a ≤ | α| a+
| β| a
Cite this article : Anh Tuan B, Anh Thi N A brief introduction to Quillen conjecture Sci Tech Dev J.;
22(2):235-238
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One can prove that two absolute values are equivalent
if there exists c > 0 such that | α|1=| α| c An
equiva-lent class of non-trivial absolute values is called a place
of K.
Definition 2 Let K be a number field Given a finite
set S of places of K that containing all archimedean places We define the ring of S-integers to be the set
O K,S={a ∈ K∥ a| v ≤ 1 for all v /∈ S}
Definition 3 Let G be an algebraic group defined over
K Then an S-arithmetic subgroup Γof G(K) is a sub-group commensurable with G(Ok,S).
Let us consider one example as follows:
Example 1 In this example, we will calculate the
possi-bilities of S-integers in case Kis the fieldQof rational
numbers Let pbe a prime, then each rational num-ber q ∈ Q ∗ can be written uniquely as q = p r m
n ′ , where (m, n) = 1and both mand nare not divisible by p The p−adic valuation v p:Q → Zis defined by v p (q) =
r and v p(0) =∞ And the p-adicabsolute value asso-ciated to the prime pis defined |q| p = p −v p (q)
Os-trowskis theorem3says that every non-trivial absolute value onQis equivalent to |·| p for some prime p ≤ ∞.
Note that we consider the usual absolute value onQas
|.|∞ , the case that p = ∞ Forconvenience we identify the absolute value v p by its related prime p Thus the set of places Sis of the form P ∪{∞}where P is a set of prime numbers.
Suppose that S = {∞, p1 , p2, , p n } then
O Q,S=ZS=Z
[ 1
p1,
1
p2, ··· , 1
p n
]
And thus we have a class of S-arithmetic groups
SL(
n,Z[1
m
])
where m is square-free positive integer.
QUILLENS CONJECTURE AND RELATED RESULTS
In this section, we will present the details of Quillens conjecture and also give a comprehensive list of the results related to this conjecture
Conjecture 12Let ℓ be a prime number Let K be a number field withζℓ ∈ K, and S a finite set of places containing the infinite places and the places over ℓ.
Then the natural inclusion O K,S , → C makes
H ∗(
GL n
(
O K,S
)
;Fℓ
)
a free module over the
coho-mology ring H ∗
cts (GL n(C);Fℓ)
This conjecture was re-stated as
Conjecture 2 H ∗(Γ,Fℓ)is a free module over the sub-ring Fℓ [c1, c2,··· ,cn], where Γ = GL n(O) and O is the ring of S−integers in a number field and c i is some Chern classes.
Over the last forty years, many efforts have been put into this conjecture and many results have been pub-lished Some of them show the positiveness of the conjecture, some disproved it Here we give a brief list of the outstanding results
Positive cases, in which the conjecture has been estab-lished, are:
1 n = ℓ = 2, O k,S=Z[1
2]by Mitchell4in 1992
2 n = 3, ℓ = 2, O k,S=Z[1
2
]
by Henn5in 1997
3 n = ℓ = 2, O k,S=Z[i,12]
by Weiss (supervised
by Henn) in 2007
4 n = ℓ = 2, O k,S=Z[√
−2,1 2
]
by Bui & Rahm6
in 2017 Despite of the above positive cases, many mathemati-cians have been trying to find a counterexample In
1998, Mitchell4show that the restriction map
H ∗(
GL n
( Z
[ 1 2
])
,F2
)
→ H ∗
(
T n
( Z
[ 1 2
])
,F2 )
from GL n
(
Z[1 2
])
to the subgroup T n
(
Z[1 2
])
of diag-onal matrices is isomorphic to
F2[w1, w2, , w n]⊗ Λ[e1 , e2, , e 2n −1] Henn, Lannes and Schwartz7show that, in the case
of the Quillen’s conjecture for GL n
(
Z[1 2
]) the above restriction map is injective This shed the light on how
to find counterexamples The non-injectivity of the restriction map has been shown in some cases:
1 n ≥ 32 and ℓ = 2 by Dwyer8
2 n ≥ 14 and ℓ = 2 by Henn and Lannes9
3 n ≥ 27 and ℓ = 3 by Anton10
QUILLEN CONJECTURE FOR FARRELL-TATE COHOMOLOGY
In effort of making a refinement of Quillens conjec-ture, Rahm and Wend, in their paper11, state that the conjecture is true for Farrel-Tate cohomology in the following cases
Theorem 111Let ℓ be a prime number Let K be a number field withζℓ ∈ K, and S a finite set of places containing the infinite places and the places over ℓ
1 The Quillen conjecture is true for
Farrell-Tate cohomology of SL2
(
OK,S
) More precisely, the natural morphismFℓ [c2] ∼ = H ·
cts (SL2(C),Fℓ)→
H ·(
SL2(
O K,S
)
,Fℓ
) extends to a morphism
ϕ : Fℓ
[
c2, c −1
2
]
→ b H ∗(
SL2(
O K,S
)
,Fℓ
)
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which makes b H •(
SL2(
O K,S
)
,Fℓ
)
a freeFℓ
[
c2, c −1
2
]
-module.
2 The Quillen conjecture is true for SL2
(
O K,S
)
in co-homological degrees above the virtual coco-homological dimension.
They also refine the conjecture to
Conjecture 3 11 Let K be a number field Fix a prime ℓ such thatζℓ ∈ K, and an integer n < ℓ As-sume that S is a set of places containing the infinite places and the places lying over ℓ If each cohomol-ogy class of GL n
(
O K,S
)
is detected on some finite sub-group, then H •
cts (GL n(C),F ℓ)is a free module over the image of the restriction map H •
cts (GL n(C),Fℓ)→
H • (GL n(C),F ℓ).
By the above theorem, Rahm and Wendt have made
the following observations for SL2 over rings of S-integers at odd prime ℓ:
1 The original Quillen conjecture holds for group
cohomology H ∗(
SL2(
O k,S
)
;Fℓ
) above the vir-tual cohomology dimension
2 The refined Quillen conjecture holds for Farrell-Tate cohomology bH ∗(
SL2(
O k,S
)
;Fℓ
)
RECENT RESULTS
In this section, we will introduce some results related
to Quillen conjecture that the first author and his col-laborator had published
Theorem 26
The cohomology ring H ∗(
SL2( Z[√ −2][1
2
])
;F2
)
is isomorphic to the free module
F2[e4] (x2, x3, y3, z3, s3, x4, s4, s5, s6)
overF2[e4](the image of H ∗
cts (SL2(C),F2)), where the subscript of the classes specifies their degree, e4 is the image of the second Chern class of the natural repre-sentation of SL2(C), and all other classes are exterior classes.
This gives one more positive example for Quillen con-jecture Reader can find the details of the computa-tion in6
CONCLUSION
Over the last forty years, many results have been pub-lished about Quillen conjecture Some give positive results, some introduce counterexamples in which the conjecture fails, and some make refinements In11, Rahm and Wendt have stated that for large primes
ℓ, we can determine precisely the module structure above the virtual cohomological dimension There-fore, the future work on this conjecture should focus
on the cases of small primes In order to refine the conjecture or find other counterexamples, there are two sources that we can look at:
• (i) The injectivity of the restriction map
H ∗(
GL n
(
Z[1 2
])
,F2
)
→ H ∗(
T n
(
Z[1 2
])
,F2 )
from GL n
(
Z[1 2
])
to the subgroup T n
(
Z[1 2 ])
of diagonal matrices
• (ii) In12, Wendt found that the image of the Quillen homomorphism is not free in the cases that he is observing
In order to examine a wide range of S-arithmetic
groups, we may need to subdivide the spaces on that those groups act to get a rigid action This problem may become a serious problem since those cell com-plexes can be very big and complicated The first au-thor and his collaborator have developed an algorithm named Rigid Facet Subdivision13 to overcome this problem
COMPETING INTERESTS
The authors declare that they have no conflicts of in-terest
AUTHORS’ CONTRIBUTIONS
Nguyen Anh Thi have collected the information of Quillen’s conjecture over
time and pointed out the two main keys to attack the conjecture by the work
of Henn and Wendt et al Bui Anh Tuan provided an
example which conrms Quillen’s conjecture
ACKNOWLEDGMENTS
The authors were funded by Vietnam National Uni-versity, Ho Chi Minh City (VNU- HCM) under grant number C2018-18-02 We would like to thank Alexander D Rahm for having supported in the de-velopment of this project
REFERENCES
1 Morris DW Introduction to Arithmetic Groups Deductive Press; .
2 Quillen D The spectrum of an equivariant cohomology ring.
I, II Ann of Math 1971;94(1):549–572 ibid (2) 94 (1971), 573-602.
3 Ostrowski A Über einige Lsungen der Funktionalgleichung Acta Mathematica (2nd ed) 1916;41:271–284 Available from: 10.1007/BF02422947.ISSN0001-5962
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6 Bui AT, Rahm AD Verification of the Quillen conjecture for Bianchi groups.; 2018 Submitted.
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